The B -> pi K Puzzle and its Relation to Rare B and K Decays

The Standard-Model interpretation of the ratios of charged and neutral B-> pi K rates, R_c and R_n, respectively, points towards a puzzling picture. Since these observables are affected significantly by colour-allowed electroweak (EW) penguins, this ``B -> pi K puzzle'' could be a manifestation of new physics in the EW penguin sector. Performing the analysis in the R_n-R_c plane, which is very suitable for monitoring various effects, we demonstrate that we may, in fact, move straightforwardly to the experimental region in this plane through an enhancement of the relevant EW penguin parameter q. We derive analytical bounds for q in terms of a quantity L, that measures the violation of the Lipkin sum rule, and point out that strong phases around 90 deg are favoured by the data, in contrast to QCD factorisation. The B -> pi K modes imply a correlation between q and the angle gamma that in the limit of negligible rescattering effects and colour suppressed EW penguins depends only on the value of L. Concentrating on a minimal flavour-violating new-physics scenario with enhanced Z^0 penguins, we find that the current experimental values on B -> X_s mu^+ mu^- require roughly L pi K data give L = 5.7 +- 2.4, L has either to move to smaller values once the B -> pi K data improve or new sources of flavour and CP violation are needed. In turn, the enhanced values of L seen in the B -> pi K data could be accompanied by enhanced branching ratios for rare decays. Most interesting turns out to be the correlation between the B -> pi K modes and BR(K^+ -> pi^+ nu nu), with the latter depending approximately on a single ``scaling'' variable \bar L= L (|V_{ub}/V_{cb}|/0.086)^2.3.Comment: 19 pages, 7 figures, a few typos corrected and two references adde

Next-to-leading order short distance QCD corrections to the effective ΔS=2\Delta S = 2 Hamiltonian, implications for the KLK_L-KSK_S mass difference

We report on the results of a calculation of next-to leading order short distance QCD corrections to the coefficient $\eta_1$ of the effective $\Delta S = 2$ Lagrangian in the standard model and discuss the uncertainties inherent in such a calculation. As a phenomenological application we comment on the contributions of short distance physics to the ${\rm K}_{\rm L}$--${\rm K}_{\rm S}$ mass difference. This report is based on research work done in collaboration with Ulrich Nierste.Comment: 4 pages and 2 figures, uses LaTeX espcrc2 documentstyle option (all necessary files included in an uuencoded compressed tar file). Invited talk at the conference 'QCD94', Montpellier, France, 7-13 July 1994, to appear in the proceedings. TUM-T31-72/9

Two--Loop Anomalous Dimension Matrix for ΔS=1\Delta S=1 Weak Non-Leptonic Decays II: O(αeαs){\cal O}(\alpha_e \alpha_s)

We calculate the $10\times 10$ two--loop anomalous dimension matrix to order \ord(\alpha_e \alpha_s) in the dimensional regularization scheme with anticommuting $\gamma_5$ (NDR) which is necessary for the extension of the $\Delta S=1$ weak Hamiltonian involving electroweak penguins beyond the leading logarithmic approximation. We demonstrate, how a direct calculation of penguin diagrams involving $\gamma_5$ in closed fermion loops can be avoided thus allowing a consistent calculation of two--loop anomalous dimensions in the simplest renormalization scheme with anticommuting $\gamma_5$ in $D$ dimensions. We give the necessary one--loop finite terms which allow to obtain the corresponding two--loop anomalous dimension matrix in the HV scheme with non--anticommuting $\gamma_5$.Comment: 25 page

Generalized Factorization in Non-leptonic Two-Body B-Decays: a Critical Look

We reanalyze critically the generalized factorization hypothesis in non-leptonic two-body B-decays discussed recently by several authors. In particular we address the determination of the factorization scale $\mu_f$ and of the non-perturbative parameters $\xi^{NF}_1 (m_b)$ and $\xi^{NF}_2 (m_b)$ which are supposed to measure non-factorizable contributions to hadronic matrix elements with $\xi^{NF}_i (\mu_f) = 0$. We emphasize that both $\mu_f$ and $\xi^{NF}_i (m_b)$ are renormalization scheme dependent and we demonstrate analytically and numerically that for any chosen scale \mu_f=\ord(m_b) it is possible to find a renormalization scheme for which $\xi^{NF}_1 (\mu_f) = \xi^{NF}_2 (\mu_f)=0$. The existing data indicate that such "factorization schemes" differ from the commonly used schemes NDR and HV. Similarly we point out that the recent extractions of the effective number of colours $N^{eff}$ from two-body non-leptonic B-decays while $\mu$ and renormalization scheme independent suffer from gauge dependences and infrared dependences.Comment: 20 pages, 2 figure

New Aspects of B -> pi pi, pi K and their Implications for Rare Decays

We analyse the B -> pi pi, pi K modes in the light of the most recent B-factory data, and obtain the following new results: (i) the B0 -> pi+ pi-, pi- K+ modes prefer gamma=(74+-6)deg, which - together with |V_ub/V_cb| - allows us to determine the ``true'' unitarity triangle and to search for CP-violating new-physics contributions to B0_d-\bar B0_d mixing; (ii) the B -> pi K puzzle reflected in particular by the low experimental value of the ratio R_n of the neutral B -> pi K rates persists and still favours new physics in the electroweak penguin sector with a new CP-violating phase phi ~ -90deg, although now also phi ~ +90deg can bring us rather close to the data; (iii) the mixing-induced B0 -> pi0 K_S CP asymmetry is a sensitive probe of the sign of this phase, and would currently favour phi ~ +90deg, as well as the direct CP asymmetry of B+- -> pi0 K+-, which suffers, however, from large hadronic uncertainties; (iv) we investigate the sensitivity of our B -> pi K analysis to large non-factorizable SU(3)-breaking effects and find that their impact is surprisingly small so that it is indeed exciting to speculate on new physics; (v) assuming that new physics enters through Z0 penguins, we study the interplay between B -> pi K and rare B, K decays and point out that the most recent B-factory constraints for the latter have interesting implications, bringing us to a few scenarios for the future evolution of the data, where also the mixing-induced CP violation in B0 -> pi0 K_S plays a prominent role.Comment: Two references added, to appear in the European Physical Journal

Completing NLO QCD Corrections for Tree Level Non-Leptonic Delta F = 1 Decays Beyond the Standard Model

In various extensions of the Standard Model (SM) tree level non-leptonic decays of hadrons receive contributions from new heavy gauge bosons and scalars. Prominent examples are the right-handed W' bosons in left-right symmetric models and charged Higgs (H^\pm) particles in models with extended scalar sector like two Higgs doublet models and supersymmetric models. Even in the case of decays with four different quark flavours involved, to which penguin operators cannot contribute, twenty linearly independent operators, instead of two in the SM, have to be considered. Anticipating the important role of such decays at the LHCb, KEKB and Super-B in Rome and having in mind future improved lattice computations, we complete the existing NLO QCD formulae for these processes by calculating O(alpha_s) corrections to matching conditions for the Wilson coefficients of all contributing operators in the NDR-\bar{MS} scheme. This allows to reduce certain unphysical scale and renormalization scheme dependences in the existing NLO calculations. Our results can also be applied to models with tree-level heavy neutral gauge boson and scalar exchanges in Delta F = 1 transitions and constitute an important part of NLO analyses of those non-leptonic decays to which also penguin operators contribute.Comment: 24 pages, 6 figure

Upper Bounds on Rare K and B Decays from Minimal Flavour Violation

We study the branching ratios of rare K and B decays in models with minimal flavour violation, using the presently available information from the universal unitarity triangle analysis and from the measurements of Br(B -> X_s gamma), Br(B -> X_s l^+l^-) and Br(K^+ -> pi^+ nu nubar). We find the following upper bounds: Br(K^+ -> pi^+ nu nubar) pi^0 nu nubar)< 4.6 10^{-11}, Br(K_L -> mu mubar)_{SD} X_s nu nubar)< 5.2 10^{-5}, Br(B -> X_d nu nubar) mu mubar)< 7.4 10^{-9}, Br(B_d -> mu mubar)< 2.2 10^{-10} at 95 % probability. We analyze in detail various possible scenarios with positive or negative interference of Standard Model and New Physics contributions, and show how an improvement of experimental data corresponding to the projected 2010 B factory integrated luminosities will allow to disentangle and test these different possibilities. Finally, anticipating that subsequently the leading role in constraining this kind of new physics will be taken over by the rare decays K^+ -> pi^+ nu nubar, K_L -> pi^0 nu nubar and B_{s,d} -> mu mubar, that are dominated by the Z^0 -penguin function C, we also present plots for several branching ratios as functions of C . We point out an interesting triple correlation between K^+ -> pi^+ nu nubar, B -> X_s gamma and B -> X_s l^+l^- present in MFV models.Comment: 26 pages, 12 figure

Two-Loop QCD Anomalous Dimensions of Flavour-Changing Four-Quark Operators Within and Beyond the Standard Model

We calculate the two-loop QCD anomalous dimension matrix (ADM) gamma^(1)_NDR in the NDR-MSbar scheme for all the flavour-changing four-quark dimension-six operators that are relevant in both the Standard Model and its extensions. Both current-current and penguin diagrams are included. Some of our NDR-MSbar results for Delta F=1 operators overlap with the previous calculations, but several others have never been published before. In the case of Delta F=2 operators, our results are compatible with the ones obtained by Ciuchini et al. in the Regularization-Independent renormalization scheme, but differ from their NDR-MSbar results. In order to explain the difference, we calculate the ADM of Delta F=2 operators again, extracting it from the ADM of Delta F=1 operators.Comment: 37 pages, 1 figur

epsilon'/epsilon in the Standard Model

In order to provide an estimate of eps'/eps several effective theories and physical effects have to be disentangled. In this talk I discuss how it is possible to predict eps '/eps taking into account all sources of large logs. The numerical result one obtains, \eps '/\eps \sim (1.7\pm 0.6) \cdot 10^{-4}, is in good agreement with present measurements.Comment: Talk presented at QCD2000, Montpellier july 200

The Rare Decays K→πννˉK\to\pi\nu\bar\nu, B→XννˉB\to X\nu\bar\nu and B→l+l−B\to l^+l^- -- An Update

We update the Standard Model predictions for the rare decays $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$. In view of improved limits on $B_s$--$\bar B_s$ mixing we derive a stringent and theoretically clean Standard Model upper limit on $B(K^+\to\pi^+\nu\bar\nu)$, which is based on the ratio of $B_d$--$\bar B_d$ to $B_s$--$\bar B_s$ mixing, $\Delta M_d/\Delta M_s$, alone. This method avoids the large hadronic uncertainties present in the usual analysis of the CKM matrix. We find $B(K^+\to\pi^+\nu\bar\nu)< 1.67\cdot 10^{-10}$, which can be further improved in the future. In addition we consider the extraction of $|V_{td}|$ from a future measurement of $B(K^+\to\pi^+\nu\bar\nu)$, discussing the various sources of uncertainties involved. We also investigate theoretically clean constraints on $B(K_L\to\pi^0\nu\bar\nu)$. We take the opportunity to review the next-to-leading order (NLO) QCD corrections to $K\to\pi\nu\bar\nu$, $K_L\to\mu^+\mu^-$, $B\to X\nu\bar\nu$ and $B\to l^+l^-$, including a small additional term that had been missed in the original publications. The phenomenological impact of this change is negligible, the corresponding numerical shift being essentially within the small perturbative uncertainties at the NLO level.Comment: 19 pages, no figure